Santi Cazorla's double ensured three points for Arsene Wenger's side, but Andreas Weimann's late strike had initially looked to have snatched a point for the struggling Villans.
"We haven't had the rub of the green of late, and when we went to 80 minutes at 1-1 I did fear we wouldn't get that goal," Jenkinson told the club's official website.
"The way we were playing and the [number of] chances we created, it would have been criminal if we didn't win the game but football doesn't always go the way it should. But we got the goal we deserved and I am delighted with it.
"In the league our form has been good, but other results haven't been what we wanted. Sometimes that happens in football and we had to react."
Arsenal face their bitter north London rivals, Tottenham, at White Hart Lane next Sunday and Jenkinson is hoping for a repeat result, with both clubs locked in a battle to secure fourth spot in the Premier League.
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"Hopefully it will give us the belief and confidence we need to go there and win.
"You want to play in all the big games and it doesn't get much bigger than playing Spurs.
"I just hope we can go there and play the way we can play and get the result I know we are capable of."